Pure Electron Plasma Research Articles

Experimental breaking of an adiabatic invariant.

When a cylindrical pure electron plasma is displaced from the center of the trap, it performs a bulk circular orbital motion known as the l=1 diocotron mode. The slow application of a perturbing potential to a patch on the trap wall distorts the orbit into a noncircular closed path. Experiments and a simple theoretical model indicate that the area by the loop is an adiabatic invariant. Detailed studies are made of the breaking of the invariant when perturbations are rapidly applied. When the perturbation is applied with discontinuous time derivatives, the invariant breaking greatly exceeds the predictions of the standard theory for smooth perturbations.

A class of coherent vortex structures in rotating non-neutral plasma

A class of nonaxisymmetric (∂/∂θ≠0) rotating equilibria is investigated theoretically for strongly magnetized, low-density (ωpe2/ωce2≪1) pure electron plasma confined radially by a uniform axial magnetic field B0ez between concentric, perfectly conducting, cylindrical walls located at radii r=rw and r=rI≤rw. The analysis is based on a nonrelativistic, guiding-center model in the cold-fluid limit that treats the electrons as a massless fluid (me→0) with E×B flow velocity Ve=−(c/B0)∇φ×ez. Assuming two-dimensional spatial variations (∂/∂z=0), the continuity-Poisson equations are analyzed for rotating coherent structures that are stationary (time independent) in a frame of reference rotating with angular velocity ωr=const about the cylinder axis (r=0). The equilibrium Poisson equation ∇2ψ=−4πe2ne(ψ)+2ωreB0/c is solved exactly for the particular case where the electron density ne(ψ) is a linear function of the streamfunction ψ=−eφ+ωr(eB0/2c)r2, and the plasma fills the region between the conducting walls, with ne=0 at r=rI and r=rw. It is found that this class of rotating equilibria can exhibit large-amplitude, nonaxisymmetric, vortex structures characterized by strong azimuthal density bunching and circulating electron flow within the density bunches. Nonlinear stability properties are investigated using the Lyapunov method, and the vortex equilibria with azimuthal mode number l=1 are shown to be stable.

Equilibration rate of spin temperature in a strongly magnetized pure electron plasma

The equilibration of spin temperature Tspin with kinetic temperature T is examined in a weakly correlated pure electron plasma in the strongly magnetized limit, where the distance of closest approach is large compared to the Larmor radius. In this limit, the spin precession frequency Ωp=gΩc/2 is large so the component of spin along the magnetic field is an adiabatic invariant that is broken only by resonant magnetic fluctuations of frequency Ωp. (Here Ωc is the electron cyclotron frequency and g≂2.002.) In this case, the most important spin flip mechanism stems from electron–electron collisions in a spatially inhomogeneous magnetic field. Such collisions cause an exchange of spin and cyclotron quanta, and consequently the conventional many-electron adiabatic invariant (i.e., the total number of cyclotron quanta) is broken and is replaced by a new adiabatic invariant, equal to the sum of the spin and cyclotron actions. A quantum Boltzmann equation is derived to describe the equilibration of Tspin toward T.

Exponential growth of an unstable l=1 diocotron mode for a hollow electron column in a warm-fluid model

Numerical investigations of a warm-fluid model with an isothermal equation of state for the perpendicular dynamics of an axisymmetric, magnetically confined pure electron plasma predict an exponentially unstable, l=1, diocotron mode for hollow density profiles. The unstable mode can be identified with a stable, nonsmooth mode that exists in cold drift models but which is destabilized by finite temperature effects. The unstable mode has many properties similar to the experimental results reported by Driscoll [Phys. Rev. Lett. 64, 645 (1990)].

Experimental dynamics of an annulus of vorticity in a pure electron plasma

The dynamics of hollow, cylindrically shaped pure electron plasmas are studied. Since these plasmas behave like two-dimensional (2-D) fluids with low viscosity, their evolution parallels the evolution of 2-D fluid vorticity annuli inside frictionless circular containers. Growth of the Kelvin–Helmholtz instability in circular geometry is observed. The formation and subsequent interaction of vortices are studied as a function of ring thickness. Small initial perturbations can dramatically increase the symmetry and repeatability of the dynamics.

Observation of the ion resonance instability.

Observation of the ion resonance instability in a pure electron plasma trap contaminated with a small population of ions is reported. The ion population is maintained by ionization of the background gas. The instability causes the plasma to move steadily off-center while undergoing [ital l]=1 diocotron oscillations. The observed scaling of the maximum growth point is presented, and the growth rate and its dependence on ion density are discussed. Several aspects of the observed behavior are not in agreement with previous theory but derive from the transitory nature of the ion population.

Coherent structures in rotating non-neutral plasma

Nonaxisymmetric (∂/∂θ≠0) rotating equilibria are investigated theoretically for strongly magnetized, low-density (ωpe2/ωce2≪1) pure electron plasmas confined in cylindrical geometry. These two-dimensional equilibria are also called rotating coherent structures, and are stationary (time independent) in a frame of reference rotating with angular velocity ωr=const about the cylinder axis (r=0). Radial confinement of the pure electron plasma is provided by a uniform axial magnetic field B0ez, and a grounded, perfectly conducting, cylindrical wall is located at radius r=rw. The analysis is based on a nonrelativistic, guiding-center model in the cold-fluid limit (the continuity and Poisson equations) that treats the electrons as a massless fluid (me→0) with E×B flow velocity Ve=−(c/B0)∇φ×ez. Within this model, general rotating equilibria with electron density ne≡nR(r,θ−ωrt) and electrostatic potential φ≡φR(r,θ−ωrt) have the property that the electron density is functionally related to the streamfunction ψR=−eφR+ωr(eB0/2c)r2 by nR=nR(ψR). The streamfunction ψR satisfies the nonlinear equilibrium equation ∇2ψR=−4πe2nR(ψR)+2ωreB0/c with ψR=ωr(eB0/2c)rw2≡ψw=const on the cylindrical wall at r=rw. Following a general discussion of rotating equilibria, an integral equation formulation of the nonlinear equilibrium equation is developed to investigate equilibria with ‘‘waterbag’’ (step-function) density profiles. In this investigation, a numerical method is formulated that can be used to construct diverse classes of highly nonlinear waterbag equilibria. This method is employed to investigate two classes of nonaxisymmetric equilibria that are nonlinear extrapolations of well-known small-amplitude equilibria. These two classes of rotating equilibria bear strong similarities to coherent structures observed experimentally by Driscoll and Fine [Phys. Fluids B 2, 1359 (1990)].

A pulsed microchannel-plate-based non-neutral plasma imaging system

The design and operation of a pure electron plasma imaging system is described. The system yields images with high resolution and a large dynamic range. A two-dimensional image of the plasma is acquired by streaming the plasma directly onto a microchannel plate and phosphor screen. Resolution of the rapidly evolving plasma requires that all the plasma strike the microchannel plate in a single charge pulse lasting less than 1 μs. The expected performance and actual operational characteristics of the system are detailed, with special attention given to the properties of a microchannel plate used in a pulsed imaging mode.

Free expansion of a pure electron plasma column.

The collective free expansion of a magnetized pure electron plasma column along the confining axial magnetic field is experimentally and theoretically investigated. A new hydrodynamic theory for nonlinear plasma wave evolution in a bounded cylinder is described which predicts the experimental results. The plasma free expansion is initially characterized by a self-similar plasma flow resulting in a perturbed density and velocity with no characteristic length scale.

Some recent results with nonneutral plasmas

Two examples of recent research using pure electron plasmas are discussed. Various phenomena of two-dimensional fluid dynamics were investigated including vortex dynamics, vortex merger, shear-induced instability, and the decay of turbulence. In a second series of experiments, the rate at which the temperatures parallel and perpendicular to a magnetic field relax to a common value has been measured over a very wide range in temperatures and magnetic fields.

Simple theory of a nonlinear diocotron mode

A simple perturbation theory of the frequencies and density distortions of the l=1 diocotron mode on pure electron plasma columns is presented. This mode is the dynamical state of an off-axis electron column which is orbiting around the containment axis at the E×B drift velocity. The theory predicts shifts in the orbit frequency and quadrupole distortions of the plasma surface, both varying as the offset squared, in good agreement with experiments. The results also apply to a two-dimensional vortex patch in an inviscid fluid with circular boundaries.

Asymmetric stable equilibria of non-neutral plasmas.

A pure electron plasma, confined within an azimuthally symmetric boundary by a coaxial magnetic field, has an equilibrium shape which is cylindrical. We apply perturbations which break the azimuthal symmetry and deform the plasma into a noncircular shape that is stationary in the laboratory frame. These asymmetric equilibria form a broad new class of stable equilibria. A theoretical model correctly predicts the plasma shapes and explains their stability.

Parallel energy analyzer for pure electron plasma devices

A technique is presented for measuring the parallel energy distribution of magnetically confined electrons in a cylindrically symmetric pure electron plasma. In essence, the technique measures how many electrons are energetic enough to escape past applied confinement potentials. The technique does not require any secondary magnetic fields. Simplified variations of the technique are also presented which can be used at the expense of some loss of information. These techniques have been successfully used in three experimental contexts.

Cyclotron resonance phenomena in a pure electron plasma

An experiment designed to elucidate the features of cyclotron resonance in a rotating cylindrical pure electron plasma column is described. The density is well below the Brillouin limit and varies with radius, as does the rotational angular velocity. Thus the steady state is not the rigid-rotor equilibrium which is frequently studied theoretically. The readily observed modes are found to have kz≊0 and m=1,2,3,4... (eikzz+im θ) with frequencies which are within a few percent of the cyclotron frequency (140 MHz). A single m=1 mode is found that is downshifted from the cyclotron frequency by an amount equal to the m=1 diocotron frequency. For each of the higher m values, bands of closely spaced discrete modes are found which are upshifted from the cyclotron frequency by the Doppler effect of rotation. The bands of discrete modes are explained as radially trapped azimuthally propagating Bernstein modes in a rotating plasma and approximate theories for these modes are outlined. By comparing the results with experiments, plasma parameters such as the ratio of Larmor radius to plasma scale length, the central rotation frequency, and the ratio of peak to average density can be inferred.

Measurement of collisional anisotropic temperature relaxation in a strongly magnetized pure electron plasma.

The rate at which the temperatures parallel and perpendicular to a magnetic field relax to a common value has been measured in a pure electron plasma. This rate was measured over the temperature range 28 to ${10}^{4}$ K and for magnetic fields in the range 30 to 60 kG. When the cyclotron radius ${\mathit{r}}_{\mathit{c}}$ is large compared to the classical distance of closest approach b, the measured rates are described by conventional scattering theory. When ${\mathit{r}}_{\mathit{c}}$/b1, the measured rate drops precipitously as ${\mathit{r}}_{\mathit{c}}$/b is decreased, in agreement with the adiabatic-invariant theory of O'Neil and Hjorth.

Cyclotron resonance in a pure electron plasma column.

We demonstrate experimentally the effects of plasma rotation and particle thermal velocities on cyclotron resonance in a long cylindrical pure electron plasma column of low density (${\mathrm{\ensuremath{\omega}}}_{\mathit{p}}^{2}$\ensuremath{\ll}${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}^{2}$). A single m=1 mode (${\mathit{e}}^{\mathit{i}\mathit{m}\mathrm{\ensuremath{\theta}}\mathrm{\ensuremath{-}}\mathit{i}\mathrm{\ensuremath{\omega}}\mathit{t}}$) is found whose frequency \ensuremath{\omega} is down-shifted from ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$=eB/${\mathit{m}}_{\mathit{e}}$ by an amount equal to the low-frequency diocotron mode frequency (〈${\mathrm{\ensuremath{\omega}}}_{\mathit{p}}^{2}$〉/2${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$). For m=2,3,4,. . . sets of modes are found which are Doppler shifted upward from ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$. We explain these as radially trapped and azimuthally propagating Bernstein modes.

The effect of a tilted magnetic field on the equilibrium of a pure electron plasma

If the magnetic field in a pure electron plasma containment device is not aligned with the axis of the conducting walls, the electrons in the device will accumulate at the ends of the plasma where the magnetic field lines come closest to the walls and the electrons bound to the field lines can be closest to their image charges. If the plasma is also offset radially from the center (as with an l=1 diocotron mode), then more density will accumulate at one end than the other. As the plasma revolves around the center, the electrons will slosh from one end to the other, creating a measurable signal. This signal has been experimentally measured and its origin verified using a three-dimensional equilibrium code. This signal can be used experimentally to align the magnetic field with the conducting walls.

Guiding center atoms: Three-body recombination in a strongly magnetized plasma

The three-body recombination rate is calculated for an ion introduced into a magnetically confined, weakly correlated, and cryogenic pure electron plasma. The plasma is strongly magnetized in the sense that the cyclotron radius for an electron rce=(kBTe/me)1/2/Ωce is small compared to the classical distance of closest approach b=e2/kBTe, where Te is the electron temperature and Ωce=eB/mec is the electron cyclotron frequency. Since the recombination rate is controlled by a kinetic bottleneck a few kBTe below ionization, the rate may be determined by considering only the initial cascade through states of electron-ion pairs with separation of order b. These pairs may be described as guiding center atoms since the dynamics is classical and treatable with the guiding center drift approximation. In this paper, an ensemble of plasmas characterized by guiding center electrons and stationary ions is described with the BBGKY hierarchy. Under the assumption of weak electron correlation, the hierarchy is reduced to a master equation. Insight to the physics of the recombination process is obtained from the variational theory of reaction rates and from an approximate Fokker–Planck analysis. The master equation is solved numerically using a Monte Carlo simulation, and the recombination rate is determined to be 0.070(10)n2eveb5 per ion, where ne is the electron density and ve=(kBTe/me)1/2 is the thermal velocity. Also determined by the numerical simulation is the transient evolution of the distribution function from a depleted potential well about the ion to its steady state.

Global vortices in rotating pure electron plasmas

Global vortex solutions in a rotating pure electron plasma column are derived within the framework of a dissipationless, cold fluid model. These solutions represent nonlinear steady state diocotron modes and may be of relevance to recent computer simulations on crossed-field electron flow and laboratory experiments on pure electron plasma columns.

Measurements of a nonlinear diocotron mode in pure electron plasmas.

Measurements of frequencies and density perturbations of the large amplitude l=1, ${\mathrm{k}}_{\mathrm{z}}$=0 diocotron mode on pure electron plasma columns are presented. This mode is a nonlinear dynamical state of an off-axis electron column which is E\ifmmode\times\else\texttimes\fi{}B drifting around the containment axis. At large amplitudes, the plasma column distorts into an oval cross section and the mode frequency shifts by an amount proportional to mode amplitude squared. The frequency shift arises because (1) the plasma is closer to its image than a linear model assumes, and (2) the shape distortion modifies the image charge distribution.